Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

2006 ◽  
Vol 116 (2) ◽  
pp. 161-173 ◽  
Author(s):  
Subhash J. Bhatt
Keyword(s):  
Crossed Product ◽  
C Algebra ◽  
Differential Structure ◽  
Fréchet Algebra ◽  
K Theory

Non-Commutative Deformations of C(T2) and K-Theory

1997 ◽  
Vol 08 (05) ◽  
pp. 555-571
Author(s):  
Cristina Cerri
Keyword(s):  
Positive Element ◽  
Crossed Product ◽  
C Algebra ◽  
Basic Properties ◽  
C Algebras ◽  
The Family ◽  
K Theory

For each α ≥ 0, let Bα be the universal C*-algebra generated by unitary elements uα, vα and a self-adjoint hα such that ||hα|| ≤ α and [Formula: see text]. In this work we prove that the family {Bα}α ∈ [0,∞[ extend the family of soft torus with the same basic properties, i.e., the field of C*-algebras {Bα}α ∈ [0,α0] is continuous and each Bα is a crossed product of a C*-algebra homotopically equivalent to C(S1) by Z. We then show that the K-groups of Bα are isomorphic to Z ⊕ Z. Applying results from the theory of rotation algebras we prove that every positive element (n,m) in K0(Bα) satisfies |m|α ≤ 2πn. It follows that these C*-algebras are not all homotopically equivalent to each other, although they have the same K-groups.

scholarly journals DENSE m-CONVEX FRÉCHET SUBALGEBRAS OF OPERATOR ALGEBRA CROSSED PRODUCTS BY LIE GROUPS

1993 ◽  
Vol 04 (04) ◽  
pp. 601-673 ◽  
Author(s):  
LARRY B. SCHWEITZER
Keyword(s):  
Continuous Functions ◽  
Crossed Product ◽  
C Algebra ◽  
Schwartz Functions ◽  
Differentiable Functions ◽  
Dense Subalgebra ◽  
Fréchet Algebra ◽  
Locally Convex ◽  
Locally Convex Algebra ◽  
Projective Completion

Let A be a dense Fréchet *-subalgebra of a C*-algebra B. (We do not require Fréchet algebras to be m-convex.) Let G be a Lie group, not necessarily connected, which acts on both A and B by *-automorphisms, and let σ be a sub-polynomial function from G to the nonnegative real numbers. If σ and the action of G on A satisfy certain simple properties, we define a dense Fréchet *-subalgebra G ⋊σ A of the crossed product L1 (G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in σ. We give conditions on σ and the action of G on A which imply the m-convexity of the dense subalgebra G ⋊σ A. A locally convex algebra is said to be m-convex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Fréchet algebra, and is useful in modern operator theory. If G acts as a transformation group on a locally compact space M, we develop a class of dense subalgebras for the crossed product L1 (G, C0 (M)), where C0 (M) denotes the continuous functions on M vanishing at infinity with the sup norm topology. We define Schwartz functions S (M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense Fréchet *-subalgebra G ⋊σ S (M) of rapidly vanishing, G-differentiable functions from G to S (M). If the reciprocal of σ is in Lp (G) for some p, we prove that our group algebras Sσ (G) are nuclear Fréchet spaces, and that G ⋊σ A is the projective completion [Formula: see text].

The cohomology and $K$-theory of commuting homeomorphisms of the Cantor set

1999 ◽  
Vol 19 (3) ◽  
pp. 611-625 ◽  
Author(s):  
ALAN FORREST ◽  
JOHN HUNTON
Keyword(s):  
Higher Order ◽  
Cantor Set ◽  
Crossed Product ◽  
Mapping Torus ◽  
C Algebra ◽  
Third Invariant ◽  
K Theory ◽  
Torsion Free ◽  
Order Continuous

Given a $\mathbb{Z}^d$ homeomorphic action, $\alpha$, on the Cantor set, $X$, we consider the higher order continuous integer valued dynamical cohomology, $H^*(X,\alpha)$. We also consider the dynamical $K$-theory of the action, the $K$-theory of the crossed product $C^*$-algebra $C(X)\times_{\alpha}\mathbb{Z}^d$. We show that these two invariants are essentially equivalent. We also show that they only take torsion free values. Our work links the two invariants via a third invariant which is based on topological complex $K$-theory evaluated on an associated mapping torus.

scholarly journals Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

10.53733/90 ◽  
2021 ◽  
Vol 52 ◽  
pp. 109-143
Author(s):  
Astrid An Huef ◽  
Marcelo Laca ◽  
Iain Raeburn
Keyword(s):  
Regular Representation ◽  
Crossed Product ◽  
Ideal Structure ◽  
Toeplitz Algebra ◽  
Natural Numbers ◽  
Kms States ◽  
C Algebra ◽  
Affine Semigroup ◽  
Natural Dynamics ◽  
The Right

We study the Toeplitz $C^*$-algebra generated by the right-regular representation of the semigroup ${\mathbb N \rtimes \mathbb N^\times}$, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. The Crisp--Laca boundary quotient is isomorphic to the $C^*$-algebra of the group ${\mathbb Q_+^\times}\!\! \ltimes \mathbb Q$. There is a natural dynamics on the right Toeplitz algebra and all its KMS states factor through the additive boundary quotient. We describe the KMS simplex for inverse temperatures greater than one.

Central Sequence Algebras of a Purely Infinite Simple C*-algebra

2004 ◽  
Vol 56 (6) ◽  
pp. 1237-1258 ◽  
Author(s):  
Akitaka Kishimoto
Keyword(s):  
Fixed Point ◽  
Partial Isometry ◽  
Characterization Result ◽  
C Algebra ◽  
Central Sequence ◽  
Sequence Algebra ◽  
K Theory

AbstractWe are concerned with a unital separable nuclear purely infinite simple C*-algebra A satisfying UCT with a Rohlin flow, as a continuation of [12]. Our first result (which is independent of the Rohlin flow) is to characterize when two central projections in A are equivalent by a central partial isometry. Our second result shows that the K-theory of the central sequence algebra A′ ∩ Aω (for an ω ∈ βN\N) and its fixed point algebra under the flow are the same (incorporating the previous result). We will also complete and supplement the characterization result of the Rohlin property for flows stated in [12].

scholarly journals K-theory for the tame C⁎-algebra of a separated graph

2015 ◽  
Vol 269 (9) ◽  
pp. 2995-3041 ◽  
Author(s):  
Pere Ara ◽  
Ruy Exel
Keyword(s):  
C Algebra ◽  
K Theory

scholarly journals C*-Algebras over Topological Spaces: Filtrated K-Theory

2012 ◽  
Vol 64 (2) ◽  
pp. 368-408 ◽  
Author(s):  
Ralf Meyer ◽  
Ryszard Nest
Keyword(s):  
Exact Sequence ◽  
Topological Space ◽  
Spectral Sequence ◽  
Topological Spaces ◽  
C Algebra ◽  
C Algebras ◽  
Finite Spaces ◽  
K Theory ◽  
Universal Coefficient Theorem ◽  
Kasparov Theory

AbstractWe define the filtrated K-theory of a C*-algebra over a finite topological spaceXand explain how to construct a spectral sequence that computes the bivariant Kasparov theory overXin terms of filtrated K-theory.For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two C*-algebras over a spaceXwith four points that have isomorphic filtrated K-theory without being KK(X)-equivalent. For this spaceX, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem.

scholarly journals Discrete product systems with twisted units

1995 ◽  
Vol 52 (2) ◽  
pp. 317-326 ◽  
Author(s):  
Marcelo Laca
Keyword(s):  
Dynamical System ◽  
Crossed Product ◽  
Type I ◽  
Crossed Products ◽  
Product System ◽  
C Algebra ◽  
Covariant Representations ◽  
Product Systems ◽  
Recent Theorem ◽  
I Factor

The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.

Sign in / Sign up

Export Citation Format

Share Document